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0.999.../Proof from 0.333...
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0.999...
Assumptions
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Decimal long division
Decimal multiplication by a small number
Proof
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Using long division,
${\frac {1}{3}}=0.333...$
Multiplying both sides by 3, we have
${\frac {3}{3}}=0.999...$
and therefore...
$1=0.999...$